The action of a subgroup G of automorphisms of a graph X is said to be 1 2 -transitive if it is vertex-and edge-but not arc-transitive. In this case the graph X is said to be (G, 1
2 )-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2 )-transitive. The 1 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length cycle C in X is a G-alternating cycle if every other vertex of C is the head and every other vertex of C is the tail of its two incident edges in the above orientation. It transpires that all G-alternating cycles in X have the same length and form a decomposition of the edge set of X (Proposition 2.4); half of this length is denoted by r G (X ) and is called the G-radius of X. Moreover, it is shown that any two adjacent G-alternating cycles of X intersect in the same number of vertices and that this number, called the G-attachment number a G (X) of X, divides 2r G (X ) (Proposition 2.6). If X is 1 2 -transitive, we let the radius and the attachment number of X be, respectively, the Aut X-radius and the Aut X-attachment number of X. The case a G (X)=2r G (X) corresponds to the graph X consisting of two G-alternating cycles with the same vertex sets and leads to an arc-transitive circulant graph (Proposition 2.4). If a G (X )=r G (X ) we say that the graph X is tightly G-attached. In particular, a 1 2 -transitive graph X of valency 4 is tightly attached if it is tightly Aut X-attached. A complete classification of tightly attached 1 2 -transitive graphs with odd radius and valency 4 is obtained (Theorem 3.4).
1998 Academic Press
1. INTRODUCTORY REMARKS
Throughout this paper graphs are simple and, unless otherwise specified, undirected and connected. Furthermore, all graphs and groups are assumed to be finite. For group-theoretic terms not defined here we refer the reader to [12,16].
If X is a graph let V(X ) and E(X ) denote the respective sets of vertices and edges. For v 1 , ..., v k # V(X) and a positive integer i we let N i (v 1 , ..., v k ) denote the set of all vertices in X at distance i from the set [v 1 , ..., v k ].